Complete Modal Decomposition for Optical Waveguides

Virtually all electromagnetic waveguiding structures support a multiplicity of modes. Nevertheless, to date, an experimental method for unique decomposition of the fields in terms of the component eigenmodes has not been realized. The fundamental problem is that all current attempts of modal decomposition do not yield phase information. Here we introduce a noninterferometric approach to achieve modal decomposition of the fields at the output of a general waveguiding structure. The technique utilizes a mapping of the two-dimensional field distribution onto the one-dimensional space of waveguide eigenmodes, together with a phase-retrieval algorithm to extract the amplitudes and phases of all the guided vectorial modes. Experimental validation is provided by using this approach to examine the interactions of 16 modes in a hollow-core photonic-band gap fiber.

Figure 1

Two distinct amplitude decompositions, in the waveguide eigenmode basis, for a far-field intensity measured at the waveguide output. Both yield a reconstruction that fits the measured intensity.

Figure 2

(a) Scanning electron microscope image of the PBG fiber cross section. Calculated intensities and vector field distributions for the (b) ${\mathrm{TE}}_{01}$, (c) ${\mathrm{HE}}_{11}$, and (d) ${\mathrm{HE}}_{21}$. All plotted with the degeneracy phases ${\varphi }_n$ set to 0.

Figure 3

Measured and reconstructed intensity distributions for the short-fiber case. The circle in the near-field images represents the location of the fiber core-cladding interface. The two circles in the far-field images represent the location of the first and second zeros of a far-field image of a uniformly distributed field having the shape and extent of the fiber core.

Figure 4

(a) Complete modal decomposition of the 16 lowest-energy modes, with angular momentum $m\le 4$ for the short-fiber case. The lower, middle, and upper panels depict the expansion coefficients modulus squared $|c_n{|}^2$ (normalized such that $\sum _{}^{}|c_n{|}^2=1$), the relative phases ${\psi }_n$, and the degeneracy phases ${\varphi }_n$, respectively. (b) The decomposition results of the long fiber (red bars) and a theoretical estimation (black bars). (c) The decomposition results of the bent fiber (red bars) and the calculated results based on a mode-coupling model (black bars). Both theoretical estimates shown in (b) and (c) use the results in (a) as initial conditions.